Seeing as his missiles obliterated the NC Percy was 'fighting', Joe just turned around and proceeded to go towards the reactor.
Slowly but steadily radiation levels increased, and reached the point at which they cause random glitches in the most sensitive electronics, mostly sensors. At this point he was close to the reactor though.
He didn't expect software hacking to work, as even the lowest security levels for reactor controls prefer a standalone system, a.k.a a system not connected to any other system and not featuring any wireless access. Sensor data might be transmitted through a separate system connected to the sensors only, but that still prevented anyone from hijacking the reactor without being physically present at the control switches. In order to shut this reactor down, Joe'd have to leave the Swarm's cockpit, which would expose him to a most likely fatal dose of radiation at this point. Secondly, since the reactor had failed, there was no guarantee the control systems could still be used to control anything.
Then he had an idea: Most commonly used reactors are controlled using control rods; rods made from a material which absorbs neutrons, slowing down or halting the nuclear reaction within the reactor. If he could somehow manually reinsert all the control rods fully into the reactor, it should shut down. If that doesn't do it, it was impossible to stop the reaction at this point, as the core had fully melted.
As he got even closer to the reactor, he could see doors having been blown off their frames by some kind of shockwave, the radiation within his cockpit steadily increased to dangerous levels. His NC's sensors and recievers were already disturbed sufficiently by the radiation to make understanding any comms impossible. The camera's were doing only slightly better, but at least their images were clear enough to see some contrasts.
Joe then stepped into the hall containing the reactor. Thankfully, the core was submerged in water, but the pressurized container containing it and the control rods had completely burst, and only the secondary pool was intact enough to contain water, although even it had several bumps and even holes in the upper parts. The exposed core caused the surrounding water to emit a blue light, as extremely fast electrons were slowed down by the currently boiling water. The upper 6th of the fuel rods of the core had partially melted at some point in time before though, exposing the blank fuel. The lid supposed to seal the pool while the reactor is under operation was nowhere to be found, it was probably completely shattered by the burst of the pressurized container. As he looked he noticed a rather unusual structure on the side of the pool, a few centimeter's above the core's top. At closer inspection it turned out to be some kind of refill mechanism, as water was constantly flowing into the pool from it. It was not a standard measure though, which concerned Joe, however at the moment it was the core's only source of coolant, keeping it from melting. It suddenly dawned on Joe what happened here.
For some reason, either intentional manipulation or a technical failure, the water within the pressurized container was no longer exchanged, causing it to heat up beyond limits and evaporate. The increased pressure was too much for the container, which lacked a failsafe valve, causing it to burst. Before that, the top of the fuel rods must've not been submerged though, causing them to partially melt. As the container burst, it sent its own fragments into the pool and lid, obliterating the latter into unrecognizability. The accompanying shockwave did the reast of the damage and probably also spread the radioactive dust of the molten fuel rod-tips around.
After the burst, the water that was left in the pool, alongside the refill submerged the core again, cooling it down again and stopping the melting.
Joe did not know whether or not his teammates were able to endure this sight while having this knowledge without breaking down.

Time was on the run though, so Joe had to quickly find a way to lower the control rods into the core and get out before the damage of the radiation was too much for his body to handle. He couldn't go down there, even in his NC. The electronics would fail in that radiation.
Against all odds due to the bad image of the cameras, he managed to find a sizable chunk of concrete, probably a fragment of the lid or some support column. He took it and threw it at the mounting of the control rods. He aimed carefully to hit it from the top as finding a similar chunk was unlikely, and he didn't want to topple the entire construct.
Luckily it was a success, and the mounting, weakened by the bursting of the pressurized container it was in, gave in, causing the control rods to fall into the core. After confirming that, he ran for his life, as he did everything in his power to limit the effects. The rods would still produce a large amount of heat over a long period of time, but at least the remaining rather safe unused fuel wasn't turned into used fuel anymore, hopefully.
As Joe returned to the entrance, his teammate's geiger-counters could tell, as increased amounts of radioactive dust had gotten on the Swarm's outer armor and hull, but at least the dose within the cockpit wasn't too dangerously high anymore.
While he opened the main door of the base, he asked his teammates: "You should all take off before I do, who knows how much dust has settled in the turbines of the jet-engines of the Swarm, and I don't want to cover any of you in a cloud of radioactive dust."
After everyone got out he closed them again, taking off as the last one. The Swarm would need to thorough cleaning after this mission, and Joe a medical check.